1. Technical Field
The disclosure relates to particle system analysis. More specifically, the disclosure relates to systems and methods that involve the use of polarization characteristics of light for analyzing particle systems that include surface facets.
2. Description of the Related Art
The polarization state of light is often described in relation to a plane of incidence defined by the vectors drawn in the direction of the incident and scattered rays. The transverse electric (TE) polarization refers to the polarization state of the light, the electric field vectors of which oscillate transverse, or perpendicular to this plane of incidence. The transverse magnetic (TM) polarization refers to the polarization state of the light, the magnetic field vectors of which oscillate transverse, or perpendicular to this plane of incidence. When a complex particle system having no preferentially oriented structure is illuminated by unpolarized light, the polarization state of the scattered light in the near-backward direction (i.e., within approximately 10 or 20 degrees of the exact backscatter direction) may be biased in favor of the TM state. Because fundamental physical processes like dipole scattering and Fresnel reflections favor the TE polarization state, this state is, perhaps, more commonly seen. A bias toward the TM-state is referred to as negative polarization and this negative polarization in the near backward-scatter (˜<10 degrees from exact back-scatter) is sometimes referred to as the “Polarization Opposition Effect” (POE). Based on symmetry considerations, we expect the polarization in the exact backscatter direction to be zero.
There appears to be two branches to the negative polarization bias or POE. One branch manifests itself as an asymmetric dip in the linear polarization state at approximately or less than one degree from the exact backscatter direction. Another phenomena occurring in the near-backward-scattering direction is the photometric opposition effect. This refers to an increase in the absolute intensity of the scattered light in the backward direction. The maximum of this increase is located in the exact backward-scattering direction. Researchers observing astronomical bodies have observed that the minima position of the POE is located approximately at the half-width of the photometric opposition effect, i.e., the peak in the total intensity in the exact backscatter direction produced by the coherent backscattering mechanism, See, V. K. Rosenbush, V. V. Avramchuk, A. E. Rosenbush, and M. I. Mishchenko, “Polarization properties of the Galilean satellites of Jupiter: observations and preliminary analysis,” Astrophys. J. 487, 402–414 (1997), which is incorporated by reference herein.
Another branch appears to be symmetric and parabolically shaped and is located at larger scattering angles, approximately but not limited to 5°–20° from the exact backward-scattering direction. It has been observed that one or both of these negative polarizations may be present in the same scattering system. In this disclosure, both branches of the negative polarization phenomena are referred to as the polarization opposition effect (POE). An accounting of the long, interwoven history of the POE will not be provided, as much of this has been provided elsewhere, for instance by Muinonen, K. Muinonen, “Coherent backscattering by solar system dust particles,” in Asteroids, Comets and Meteors, ed. by A. Milani, M. Di Martino, and A. Cellino (Kluwer, Dordrecht, 1974) 271–296, and Yu. Shkuratov, A. Ovcharenko, E. Zubko, V. Kaydash, D. Stankevich, V. Omelchencko, O. Miloslavaskaya, K. Muinonen, J. Piironen, S. Kaasalainen, R. Nelson, W. Smythe, V. Rosenbush and P. Helfenstein, “The opposition effect and negative polarization of structural analogs of planetary regoliths,” Icarus 159, 396–416 (2002), each of which is incorporated herein by reference.
The asymmetric branch of the POE has been inextricably linked with another phenomena, coherent backscattering enhancement. M. I. Mishechenko, “On the nature of the polarization opposition effect exhibited by Saturn's rings,” Astrophys. J. 411, 351, 361, which is incorporated herein by reference. When a light ray traverses through a random medium to a detector, it is accompanied by another ray striking all the elements of the system in reverse order. When the detector is in the exact backscatter direction, these rays constructively interfere, because they have traversed the same pathlength but in reverse directions. When the detector is not in the exact backscatter direction, the pathlengths of these two reciprocal rays are no longer identical and we see the intensity drop off because the constructive interference condition is not optimized. The width of this peak is inversely proportional to the difference in pathlength between these two rays.
In a flourish of research along parallel lines, various researchers were able to show that the coherent backscatter mechanism was also responsible for the asymmetric branch of the POE. Earlier approaches considered rigorous methods applicable to specific scattering systems, like the use of the vector theory of coherent backscattering for a semi-infinite medium of Rayleigh particles, (See, V. D. Ozrin, “Exact solution for the coherent backscattering of polarized light from a random medium of Rayleigh scatterers,” Waves Random Media 2, 141–164 (1992); M. I. Mishchenko, “Polarization effects in weak localization of light: Calculation of the copolarized and depolarized backscattering enhancement factors,” Phys. Rev. B 44, 12, 579–12, 600 (1991); M. I. Mishchenko, “Enhanced backscattering of polarized light from discrete random media,” J. Opt. Soc. Am. A 9, 978–982 (1992); and M. I. Mishchenko, J.-M. Luck, and T. M. Nieuwenhuizen, “Full angular profile of the coherent polarization opposition effect,” J. Opt. Soc. Am. A 176, 888–891 (2000), each of which is incorporated by reference herein), or a scattering from small particles near a surface, (See, I. V. Lindell, A. H. Sihovla, K. O. Muinonen, and P. W. Barber, “Scattering by a small object close to an interface. I. Exact image theory formulation,” J. Opt. Soc. Am. A 8, 472–476 (1991); and K. O. Muinonen, A. H. Sihvola, I. V. Lindell, and K. A. Lumme, “Scattering by a small object close to an interface. I. Study of backscattering,” J. Opt. Soc. Am. A 8, 477–482 (1991), each of which is incorporated by reference herein).
More recent approaches applied Monte-Carlo-type ray-tracing computations for a generated particle system, keeping track of the phase of the scattered rays (K. Muinonen, “Coherent backscattering by absorbing and scattering media,” in Light Scattering by Nonshperical Particles, B. Gustafson, L. Kolkolova, and G. Videen, eds. (U.S. Army Research Laboratory, Adelphi, Md., 2002) 223–226, which is incorporated herein reference herein) or involved an approximate expression for the scattering by a population of scatters. See, Yu. Shkuratov, M. Kreslavsky, A. Ovcharenko, D. Stankevich, E. Zubko, C. Pieters, and G. Arnold, “Opposition effect from Clementine data and mechanisms of backscatter,” Icarus 141, 132–155 (1999), which is also incorporated by reference herein. Although the theory developed by Mishchenko is valid for arbitrary scatters, it applies only to the exact backscattering direction and to the incoherent background and thus does not describe explicitly the POE. Although these methods are significantly different in approach, there are significant similarities in their results. Part of the reason for this is that the polarization state of light scattered from a Rayleigh particle and from a surface facet are similar.
FIG. 1 shows the polarizing properties of a Rayleigh particle and surface facets as a function of scattering angle. The scattered light from the surface facet is predicted solely from the Fresnel reflection coefficients. Specifically, polarization properties of a Raleigh scatterer and of large facets, who scatter into the far field is calculated by a single Fresnel reflection, are shown. Zero degrees is the forward scatter, measured from the specular direction, and 180° is the back-scatter direction. Although the polarization properties of a Rayleigh sphere are independent of material, those calculated with Fresnel reflections are dependent on material properties of the crystal. The refractive indices used in the simulation of FIG. 1 are mwater=1.33+10−5i and maluminum=0.5+5.0i.
Because the two branches of the POE have significantly different shapes and one or both may be present in a scattering system, there is some debate as to the underlying physical mechanism of the symmetric, parabolically shaped branch. This was the subject of several discussions at a North Atlantic Treaty Organization Advance Research Workshop on the “Optics of Cosmic Dust” held din Bratislava, Slovakia, 16–19 Nov. 2001. Some researchers argued that the coherent backscatter mechanism can explain the observations of both asymmetric and symmetric branches of the POE, whereas others did not believe that the evidence as yet presented provides an adequate proof of the mechanism. It appears that the source of some of the confusion is that, in the calculations of M. I. Mishchenko, J.-M. Luck, and T. M. Nieuwenhuizen, the POEs for a population of Rayleigh scatterers have the same asymmetric shape regardless of the mean free path. In fact, the polarization is even plotted as a function of a dimensionless parameter q=klγ, where l is the mean free path (or average distance a ray will travel before interacting with another surface facet), k is the spatial frequency defined in terms of wavelength λ as k=2π/λ, and γ is the scattering angle measured from the backscatter direction; hence the angular minimum can be found directly from this plot as       q    min    ≈            1.68      kl        .  
Other methodologies that are able to produce a more symmetric, parabolic branch at larger scattering angles are either approximate techniques or ones in which the physical mechanisms are not as transparent. The presence of sometimes one and sometimes both branches measured from the same object has served to add to the confusion. Adding fuel to the fire is the experimental research of Geake and Geake. See J. E. Geake and M. Geake, “A remote-sensing method for sub-wavelength grains on planetary surfaces by optical polarimetry,” Mon. Notes R. Astron. Soc. 245, 46–55 (1990), which is incorporated herein by reference. They discovered that the angular minimal positions measured from the backscatter of their samples increase with particle size parameter. This is the opposite of what would be predicted by a coherent backscatter mechanism. Much of the current understanding of the POE is contained in four chapters (astronomical observations, laboratory measurements, theory, and numerical techniques) of a book written by participants of the aforementioned workshop; specifically, G. Videen and M. Kocifaj, eds., Optics of Cosmic Dust (Kluwer Academic, Dordrecht, The Netherlands, 2002), which is incorporated by reference herein.
One would expect that attempts have been made to explain the POE using a ray-tracing model reflecting off randomly oriented facets, and indeed this is the case. Wolff did a similar analysis over a quarter of a century ago, M. Wolff, “Polarization of light reflected from rough planetary surface,” Appl. Opt. 14, 1395–1405 (1975), which is incorporated by reference herein, and this is a starting point. Wolff derived the polarization state of light considering both single and double reflections off facets. He found that negative polarizations only occur for multiple reflections when the vector connecting the facets is oriented perpendicular to the plane of incidence. Essentially, the planes of incidence of each of the individual reflections is nearly perpendicular to the plane of incidence of the scattering system. The shadowing factor introduced in Wolff's work selectively enhances this particular component to produce the necessary negative polarization. It is important to remember that Wolff's paper precedes the rapid development of the understanding of coherent backscatter phenomena that occurred in the 1980s, and so this effect is not included in Wolff's work. As it happens, in the two-reflection analysis, Wolff considered the intensities of the rays reflecting off the facets, rather than the electric field amplitudes. He therefore did not have a mechanism to include coherent backscattering in his formulation.
Other facet models have employed the coherent backscattering mechanisms, but have treated specific scattering systems. Shkuratov considered two pairs of facets and derived approximate analytical formulae for the backscattering polarization, Yu. Shkuratov, “New mechanism of the negative polarization of light scattered by atmosphereless cosmic bodes,” Astronmicheskii Vestnik 23, 176–180 (1989), which is incorporated by reference herein. Muinonen used a facet analysis to analyze the backscattering polarization from a pair of spheres, K. Muinonen, “Scattering of light by solar system dust: the coherent backscatter phenomenon,” in 1990 Proc. of the Finnish Astron. Soc. 12, which is also incorporated herein by reference.